Chapter 19: Inference for a Single Mean

Overview

  • 19.1 Bootstrap confidence interval for a mean
    • 19.1.1 Observed data
    • 19.1.2 Variability of the statistic
    • 19.1.3 Bootstrap SE confidence interval
    • 19.1.4 Bootstrap percentile confidence interval for a standard deviation
    • 19.1.5 Bootstrapping is not a solution to small sample sizes!
  • 19.2 Mathematical model for a mean
    • 19.2.1 Mathematical distribution of the sample mean
    • 19.2.2 Evaluating the two conditions required for modeling ¯x
    • 19.2.3 Introducing the t-distribution
    • 19.2.4 One sample t-intervals
    • 19.2.5 One sample t-tests

19.1 Bootstrap confidence interval for a mean

  • We use bootstrapping to estimate the sampling distribution of a statistic.
  • The process of bootstrapping for a sample mean is the same as bootstrapping for a sample proportion.
  • Since our data is numeric, we calculate the sample mean \(\bar{x}\), instead of the sample proportion \(\hat{p}\).

19.1.1 Observed data

19.1.2 Variability of the statistic

19.1.3 Bootstrap SE confidence interval

Once we have a bootstrapped sampling distribution, we can use it to make confidence intervals.

  • The 95% percentile confidence interval is the the interval from the 2.5%ile to the 97.5%ile.
  • The 95% SE confidence interval is the quick approximation \(\mbox{point estimate} \pm 2 \cdot SE_{BS}\)
    • We are using 2 instead of 1.96.

19.1.5 Bootstrapping is not a solution to small sample sizes!

  • Note: Bootstrapping (and other statistical models) work best for larger samples.
  • The car price example had a sample size of \(n=5\), which is pretty small.

Group Activity

price
18300
20100
9600
10700
27000

  1. Open the Bootstrapping Applet. Clear the preloaded data and paste in the above price data (including the header).

  2. Check Show Sampling Options and make a bootstrap distribution of sample means from this sample.

  3. Use the SD from the distribution to calculate a 95% SE confidence interval for the mean price.

  4. Select Beyond from the count samples dropdown. Use trial and error to find the upper endpoint of a 95% percentile confidence interval. (You want the total area in both tails to be 0.05.)

19.2 Mathematical model for a mean

19.2.1 Mathematical distribution of the sample mean

19.2.2 Evaluating the two conditions required for modeling ¯x

19.2.3 Introducing the t-distribution

19.2.4 One sample t-intervals

19.2.5 One sample t-tests